Flanker Tom Johnson scored Exeter's only try of the match
Fly-half Gareth Steenson kicked 17 points as Exeter produced a spirited display to defeat a hard-working Newcastle side at Sandy Park.
A tense first-half saw both sides swap penalties and converted tries to make it 10-10 at the break.
Steenson converted a further two place kicks shortly after the restart, before Falcons centre Rob Vickerman went over to give his side the lead.
But Steenson settled Exeter's nerves with two further penalties.
In a tight opening spell of a must-win game for both sides it was the Chiefs who struck first as Steenson successfully kicked an early penalty.
The home side continued to enjoy the majority of possession but, against the run of play, Newcastle's Charlie Amesbury sliced through to score the game's first try before Jimmy Gopperth added the conversion.
Exeter struck back through a converted try from flanker Tom Johnson, but the home side's discipline faltered as Newcastle increased their intensity.
Gopperth punished the Chiefs with a second penalty to bring the scores level at the break.
Rob Baxter's men came out with renewed energy in the second half, and were rewarded with two penalties, which Steenson duly slotted over.
But the Falcons refused to buckle and poor defending from the home side enabled Vickerman to go over for his side's second try of the afternoon - converted by Gopperth to go 17-16 in front.
Steenson then took over as the Chiefs started to pile on pressure.
The ever-reliable Irishman was on hand to kick the two penalties that put his side five points clear and give the Premiership newcomers their second win in three games.
Exeter Chiefs captain Tom Hayes told BBC Radio Devon: "Eight points out of three matches - a lot of people didn't expect us to get that, but we're certainly happy that we have it.
"We came from behind a couple of times in the match to claw back the lead, and I think that showed a determination and a lot of composure on our part.
"Gareth (Steenson) is deadly accurate with the kicking, so we're certainly happy that we have him to rely on."
Newcastle head coach Alan Tait said: "It was just little things that changed the game like the penalty count, especially in the second half. It was eight penalties to one after the break and most of them were for not rolling away.
"It has happened in both our away games so we'll need to look at ourselves and ask the referee for a bit of advice and see what they say.
"But it is probably the performance that was the most upsetting. We scored two tries but we can play better than that. Exeter ran a lot straighter and harder."
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